University of Georgia, Summer 2003
Summary
The VIGRE Research Experience for Undergraduates in Mathematics at UGA will run for five weeks, June 9 - July 11, 2003. Students will work in teams guided by faculty mentors. Research teams will meet approximately 30 hours a week. The research experience is open to all majors, and it is not restricted to University of Georgia students.
Twelve $2250 stipends will be awarded to United States citizens and permanent residents. Participants may register to receive course credit. Students are responsible for the cost of tuition and fees. In addition to the stipend, we will pay for dormitory rooms for students who choose to stay in dorms. Travel expense money is available for non-UGA students.
A final research report is required of all students.
This program is supported by a VIGRE grant from the National Science Foundation and by the Franklin College of Arts and Sciences of the University of Georgia.
Research topics
Analysis on Metrized Graphs
This project will introduce participants to a variety of interrelated topics in analysis, topology, probability, and physics. The basic object of study will be a "metrized graph," which is a finite graph G equipped with parametrizations of its edges. Such graphs serve as an accessible place to model general theories in higher dimensions, but also have many unique and interesting features in their own right. We will view a metrized graph in several ways: as an electric circuit, with the edges viewed as resistors; as the setting for a random walk; and as a 1-dimensional analogue of a compact Riemann surface. We will touch on a number of ideas, including Kirchoff's laws, homology and cohomology, Polya's theorem on random walks, potential theory, and generalized Fourier series. We will also consider some open problems regarding the Laplacian on a metrized graph.
Massively parallel computation of geometrically optimal knots
How much rope does it take to tie a knot? In particular, how much rope does it take to tie a really big knot? The study of knots has recently been expanded by the appearance of "geometric knot theory"; a new field which measures the topological complexity of knotted curves in space by studying geometric "energy functionals" defined on space curves.
One classical example of such a functional is the "crossing number" of a knot; that is, the minimum number of crossings that one sees when looking at any configuration of the knot from any direction. For example, the crossing number of the "overhand knot" or trefoil knot, is known to be 3, the lowest possible crossing number for any nontrivial knot.
It would be very interesting to relate the growth rates of crossing number and the minimum amount of (unit diameter) rope required to tie a knot. For instance, it is known that any knot can be constructed with a length of rope proportional to the square of the crossing number of the knot, but it is not known whether any family of knots require more rope than a linear function of their crossing numbers.
Answering questions like this will require ropelength data for many thousands of knots with various crossing numbers. To gather this data, the geometric knot theory summer VIGRE project will work on a very large-scale parallel computation of approximately ropelength-minimizing knots, using a genetic algorithm.
The (mathematical) prerequisites for the group are very reasonable; vector calculus and some linear algebra are all that is required. However, strong programming skills are essential. One of the main topics of the summer research group will be to design a program for recognizing knots from an algebraic invariant called the HOMFLY polynomal -- in this part of the program strong modern algebra skills will be very helpful.
Program details
Schedule and requirements
The program will run during UGA's summer semester. Research teams will meet approximately 30 hours a week. Participants may not enroll for other classes or hold other employment during the period of the program. Every student must submit a final report.
Financial support
Twelve stipends of $2250 will be awarded to students who are United States citizens or permanent residents. An additional housing allowance is available for students who stay in UGA dormitories. Travel expense money is available for non-UGA students. Every qualified applicant will automatically be considered for a stipend award as well as for housing and travel reimbursements.
Other sources of financial support are available through the UGA SURP and CURO programs and from mathematics faculty research grants.
Housing and meals
Students can arrange to stay in a double occupancy room in a campus residence hall. A Monday through Friday meal plan is also available in the summer for about $16 a day. Program participants are responsible for the cost of their meals. Please contact Jennifer Peterson if you would like assistance in arranging these details.
Course credit
Program participants have the option of receiving 3 semester hours of college credit for the VIGRE Research Experience by registering for MATH 4950, Research in Mathematics (Short Session I, Summer Semester 2003). Students who register for course credit are responsible for the cost of tuition and fees (UGA Bulletin).
If you are not already a student at UGA, you must apply for transient admission (UGA Undergraduate Admissions) to receive college credit. Transient students for the summer semester should apply for admission by April 1.
Application
Deadline March 1, 2003
Required documents
Mailing address
VIGRE Summer Program
Department of Mathematics
University of Georgia
Athens GA 30602-7403
Contact information
Dr. Clint McCrory
VIGRE Director
clint@math.uga.edu
(706) 542-2576Jennifer Peterson
Educational Program Specialist
jennifer@math.uga.edu
(706) 542-2577
This page was created on December 1, 2002. It was last modified on January 3, 2003.